A055462 -id:A055462 - cp's OEIS frontend

A193430 Primes p such that p+1 is in A055462.

23, 6911, 5944066965503999

Offset: 1

Kausthub Gudipati, Jul 28 2011

Prime numbers that are one less than the product of first n superfactorials.
The next term, if it exists, has more than 75,000 digits.
Subsequence of A238265.

			23 is a term because 23 = 1! * (1! * 2!) * (1! * 2! * 3!) - 1.

Cf. A005462, A000040.

t=1;k=1;for(n=2,100,t*=n!;k*=t;if(ispseudoprime(k-1),print1(k-1", "))) \\ Charles R Greathouse IV, Jul 28 2011

a(n) are the prime values of 1! * (1! * 2!) * (1! * 2! * 3!) * (1! * 2! * 3! * ... * n!) - 1

A238265 Primes p such that p + 1 or p - 1 is in A055462.

Original entry on oeis.org

2, 3, 23, 6911, 238878721, 5944066965503999

Offset: 1

Arkadiusz Wesolowski, Feb 21 2014

Primes which are within 1 of a superduperfactorial number.

Supersequence of A193430.

			A000142(2)^3*A000142(3)^2*A000142(4) - 1 = 6911, which is prime.

Cf. A000142, A055462.

t=1; p=1; for(n=1, 6, t*=n!; p*=t; if(isprime(p-1), print1(p-1, ", ")); if(isprime(p+1), print1(p+1, ", ")));

a(n) are the prime values of 1! * (1! * 2!) * (1! * 2! * 3!) * (1! * 2! * 3! * n!) +/- 1.

A000178 Superfactorials: product of first n factorials.

Original entry on oeis.org

1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, 792786697595796795607377086400871488552960000000000000

Offset: 0

N. J. A. Sloane

a(n) is also the Vandermonde determinant of the numbers 1,2,...,(n+1), i.e., the determinant of the (n+1) X (n+1) matrix A with A[i,j] = i^j, 1 <= i <= n+1, 0 <= j <= n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
a(n) = (1/n!) * D(n) where D(n) is the determinant of order n in which the (i,j)-th element is i^j. - Amarnath Murthy, Jan 02 2002
Determinant of S_n where S_n is the n X n matrix S_n(i,j) = Sum_{d|i} d^j. - Benoit Cloitre, May 19 2002
Appears to be det(M_n) where M_n is the n X n matrix with m(i,j) = J_j(i) where J_k(n) denote the Jordan function of row k, column n (cf. A059380(m)). - Benoit Cloitre, May 19 2002
a(2n+1) = 1, 12, 34560, 125411328000, ... is the Hankel transform of A000182 (tangent numbers) = 1, 2, 16, 272, 7936, ...; example: det([1, 2, 16, 272; 2, 16, 272, 7936; 16, 272, 7936, 353792; 272, 7936, 353792, 22368256]) = 125411328000. - Philippe Deléham, Mar 07 2004
Determinant of the (n+1) X (n+1) matrix whose i-th row consists of terms 1 to n+1 of the Lucas sequence U(i,Q), for any Q. When Q=0, the Vandermonde matrix is obtained. - T. D. Noe, Aug 21 2004
Determinant of the (n+1) X (n+1) matrix A whose elements are A(i,j) = B(i+j) for 0 <= i,j <= n, where B(k) is the k-th Bell number, A000110(k) [I. Mezo, JIS 14 (2011) # 11.1.1]. - T. D. Noe, Dec 04 2004
The Hankel transform of the sequence A090365 is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - Philippe Deléham, Mar 02 2005
Theorem 1.3, page 2, of Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6, provides an example of an Abelian quotient group of order (n-1) superfactorial, for each positive integer n. The quotient is obtained from sequences of polynomial values. - E. F. Cornelius, Jr. (efcornelius(AT)comcast.net), Apr 09 2007
Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000292. seq(mul(mul(i,i=alpha..k), k=alpha..n),n=alpha..12). - Peter Luschny, Jul 14 2009
For n>0, a(n) is also the determinant of S_n where S_n is the n X n matrix, indexed from 1, S_n(i,j)=sigma_i(j), where sigma_k(n) is the generalized divisor sigma function: A000203 is sigma_1(n). - Enrique Pérez Herrero, Jun 21 2010
a(n) is the multiplicative Wiener index of the (n+1)-vertex path. Example: a(4)=288 because in the path on 5 vertices there are 3 distances equal to 2, 2 distances equal to 3, and 1 distance equal to 4 (2*2*2*3*3*4=288). See p. 115 of the Gutman et al. reference. - Emeric Deutsch, Sep 21 2011
a(n-1) = Product_{j=1..n-1} j! = V(n) = Product_{1 <= i < j <= n} (j - i) (a Vandermondian V(n), see the Ahmed Fares May 06 2001 comment above), n >= 1, is in fact the determinant of any n X n matrix M(n) with entries M(n;i,j) = p(j-1,x = i), 1 <= i, j <= n, where p(m,x), m >= 0, are monic polynomials of exact degree m with p(0,x) = 1. This is a special x[i] = i choice in a general theorem given in Vein-Dale, p. 59 (written for the transposed matrix M(n;j,x_i) = p(i-1,x_j) = P_i(x_j) in Vein-Dale, and there a_{k,k} = 1, for k=1..n). See the Aug 26 2013 comment under A049310, where p(n,x) = S(n,x) (Chebyshev S). - Wolfdieter Lang, Aug 27 2013
a(n) is the number of monotonic magmas on n elements labeled 1..n with a symmetric multiplication table. I.e., Product(i,j) >= max(i,j); Product(i,j) = Product(j,i). - Chad Brewbaker, Nov 03 2013
The product of the pairwise differences of n+1 integers is a multiple of a(n) [and this does not hold for any k > a(n)]. - Charles R Greathouse IV, Aug 15 2014
a(n) is the determinant of the (n+1) X (n+1) matrix M with M(i,j) = (n+j-1)!/(n+j-i)!, 1 <= i <= n+1, 1 <= j <= n+1. - Stoyan Apostolov, Aug 26 2014
All terms are in A064807 and all terms after a(2) are in A005101. - Ivan N. Ianakiev, Sep 02 2016
Empirical: a(n-1) is the determinant of order n in which the (i,j)-th entry is the (j-1)-th derivative of x^(x+i-1) evaluated at x=1. - John M. Campbell, Dec 13 2016
Empirical: If f(x) is a smooth, real-valued function on an open neighborhood of 0 such that f(0)=1, then a(n) is the determinant of order n+1 in which the (i,j)-th entry is the (j-1)-th derivative of f(x)/((1-x)^(i-1)) evaluated at x=0. - John M. Campbell, Dec 27 2016
Also the automorphism group order of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Is the zigzag Hankel transform of A000182. That is, a(2*n+1) is the Hankel transform of A000182 and a(2*n+2) is the Hankel transform of A000182(n+1). - Michael Somos, Mar 11 2020
Except for n = 0, 1, superfactorial a(n) is never a square (proof in link Mabry and Cormick, FFF 4 p. 349); however, when k belongs to A349079 (see for further information), there exists m, 1 <= m <= k such that a(k) / m! is a square. - Bernard Schott, Nov 29 2021

			a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |
a(7) = n! * a(n-1) = 7! * 24883200 = 125411328000.
a(12) = 1! * 2! * 3! * 4! * 5! * 6! * 7! * 8! * 9! * 10! * 11! * 12!
= 1^12 * 2^11 * 3^10 * 4^9 * 5^8 * 6^7 * 7^6 * 8^5 * 9^4 * 10^3 * 11^2 * 12^1
= 2^56 * 3^26 * 5^11 * 7^6 * 11^2.
G.f. = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

Boris Hostnik, Table of n, a(n) for n = 0..46
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Andreas B. G. Blobel, On convolution powers of 1/x, arXiv:2203.09519 [math.CO], 2022.
E. F. Cornelius, Jr. and Phill Schultz, Polynomial points , Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
Selden Crary, Factorization of the Determinant of the Gaussian-Covariance Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality, arXiv preprint arXiv:1406.6326 [math.ST], 2014-2019.
N. Destainville, R. Mosseri and F. Bailly, Configurational Entropy of Codimension-One Tilings and Directed Membranes, J. Stat. Phys. 87, Nos 3/4, 697 (1997).
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7.
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Ivan Gutman, Wolfgang Linert, István Lukovits and Željko Tomović, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., Vol. 40, No. 1 (2000), pp. 113-116.
Brady Haran and Sophie Maclean, What's special about 288?, Numberphile video (2023).
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Nick Hobson, Python program for this sequence.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Pavel L. Krapivsky, Jean-Marc Luck and Kirone Mallick, Quantum return probability of a system of N non-interacting lattice fermions, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 2 (2018), 023104; arXiv preprint, arXiv:1710.08178 [cond-mat.mes-hall], 2017-2018.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Mogens Esrom Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009.
Rémy Mosseri and Francis Bailly, Configurational Entropy in Octagonal Tiling Models, Int. J. Mod. Phys. B, Vol. 7, No. 6-7 (1993), pp. 1427-1436.
Rémy Mosseri, F. Bailly and C. Sire, Configurational Entropy in Random Tiling Models, J. Non-Cryst. Solids, Vol. 153-154 (1993), pp. 201-204.
Amarnath Murthy, Miscellaneous Results and Theorems on Smarandache terms and factor partitions, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 3.14.
Christian Radoux, Query 145, Notices Amer. Math. Soc., 25-3 (1978), p. 197.
Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Michel Waldschmidt, Schanuel Property for Elliptic and Quasi-Elliptic Functions, arXiv:2504.14041 [math.NT], 2025. Mentions this sequence, see p. 10.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Bell Number.
Eric Weisstein's World of Mathematics, Factorial Products.
Eric Weisstein's World of Mathematics, Graph Automorphism.
Eric Weisstein's World of Mathematics, Lucas Sequence.
Eric Weisstein's World of Mathematics, Superfactorial.
Eric Weisstein's World of Mathematics, Vandermonde Determinant.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads and other Mathematical competitions.

Partial products of A000142.
Cf. A002109, A036561, A000292, A098694, A098695, A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, A113535, A113153, A113154, A113122, A114045, A055462, A137986, A137987.
A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494 and A045379. - John W. Layman, Jul 28 2000
Cf. A255322, A255358, A255359, A255360.
Cf. A051675, A255321, A255323, A255344, A287013.
Cf. A348692, A349079.

[&*[Factorial(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Mar 11 2015

A000178 := proc(n)
    mul(i!,i=1..n) ;
end proc:
seq(A000178(n),n=0..10) ; # R. J. Mathar, Oct 30 2015

a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] (* Stefan Steinerberger, Mar 10 2006 *)
Table[BarnesG[n], {n, 2, 14}] (* Zerinvary Lajos, Jul 16 2009 *)
FoldList[Times,1,Range[20]!] (* Harvey P. Dale, Mar 25 2011 *)
RecurrenceTable[{a[n] == n! a[n - 1], a[0] == 1}, a, {n, 0, 12}] (* Ray Chandler, Jul 30 2015 *)
BarnesG[Range[2, 20]] (* Eric W. Weisstein, Jul 14 2017 *)

A000178(n):=prod(k!,k,0,n)$ makelist(A000178(n),n,0,30); /* Martin Ettl, Oct 23 2012 */

A000178(n)=prod(k=2,n,k!) \\ M. F. Hasler, Sep 02 2007

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j!*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

for(j=1,13, print1(prod(k=1,j,k^(j-k)),", ")) \\ Hugo Pfoertner, Apr 09 2020

A000178_list, n, m = [1], 1,1
for i in range(1,100):
    m *= i
    n *= m
    A000178_list.append(n) # Chai Wah Wu, Aug 21 2015

from math import prod
def A000178(n): return prod(i**(n-i+1) for i in range(2,n+1)) # Chai Wah Wu, Nov 26 2023

def mono_choices(a,b,n)
    n - [a,b].max
end
def comm_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        i.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
    puts comm_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

a(0) = 1, a(n) = n!*a(n-1). - Lee Hae-hwang, May 13 2003, corrected by Ilya Gutkovskiy, Jul 30 2016
a(0) = 1, a(n) = 1^n * 2^(n-1) * 3^(n-2) * ... * n = Product_{r=1..n} r^(n-r+1). - Amarnath Murthy, Dec 12 2003 [Formula corrected by Derek Orr, Jul 27 2014]
a(n) = sqrt(A055209(n)). - Philippe Deléham, Mar 07 2004
a(n) = Product_{i=1..n} Product_{j=0..i-1} (i-j). - Paul Barry, Aug 02 2008
log a(n) = 0.5*n^2*log n - 0.75*n^2 + O(n*log n). - Charles R Greathouse IV, Jan 13 2012
Asymptotic: a(n) ~ exp(zeta'(-1) - 3/4 - (3/4)*n^2 - (3/2)*n)*(2*Pi)^(1/2 + (1/2)*n)*(n+1)^((1/2)*n^2 + n + 5/12). For example, a(100) is approx. 0.270317...*10^6941. (See A213080.) - Peter Luschny, Jun 23 2012
G.f.: 1 + x/(U(0) - x) where U(k) = 1 + x*(k+1)! - x*(k+2)!/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 + 1/((k+1)!*x*G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k!*x). - Paul D. Hanna, Oct 02 2013
A203227(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 30 2016
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) = G(n+2), where G(n) is the Barnes G-function.
a(n) ~ exp(1/12 - n*(3*n+4)/4)*n^(n*(n+2)/2 + 5/12)*(2*Pi)^((n+1)/2)/A, where A is the Glaisher-Kinkelin constant (A074962).
Sum_{n>=0} (-1)^n/a(n) = A137986. (End)
0 = a(n)*a(n+2)^3 + a(n+1)^2*a(n+2)^2 - a(n+1)^3*a(n+3) for all n in Z (if a(-1)=1). - Michael Somos, Mar 11 2020
Sum_{n>=0} 1/a(n) = A287013 = 1/A137987. - Amiram Eldar, Nov 19 2020
a(n) = Wronskian(1, x, x^2, ..., x^n). - Mohammed Yaseen, Aug 01 2023
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Sum_{k=1..n} (Integral_{x=1..k+1} Psi(x) dx)).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + x*Psi(x)) dx).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + (n+1)*Psi(x) - log(Gamma(x))) dx).
Psi(x) is the digamma function. (End)

A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

Original entry on oeis.org

1, 0, 3, 0, 9, 1, 6, 7, 5, 2, 1, 9, 7, 3, 9, 2, 1, 1, 4, 1, 9, 3, 3, 1, 3, 0, 9, 6, 4, 6, 6, 9, 4, 2, 2, 9, 0, 6, 3, 3, 1, 9, 4, 3, 0, 6, 4, 0, 3, 4, 8, 7, 0, 6, 0, 2, 2, 7, 2, 6, 1, 7, 4, 1, 1, 4, 5, 1, 6, 6, 0, 6, 6, 9, 7, 8, 2, 9, 0, 4, 0, 5, 2, 9, 2, 9, 3, 1, 3, 6, 2, 5, 5, 4, 8, 0, 8, 8, 5

Offset: 1

Jean-François Alcover, Jun 02 2014

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016

			1.03091675219739211419331309646694229...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 1..2002
J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.
K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Cf. A051675, A055462, A240966, A255269.
Cf. A019727, A074962, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Cf. A002117 (zeta(3)).

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(0) = sqrt(2*Pi) (A019727),
A(1) = A = Glaisher-Kinkelin constant (A074962),
A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).
Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015

A255269 a(n) = Product_{k=1..n} k!^k.

Original entry on oeis.org

1, 4, 864, 286654464, 7132880358604800000, 993710590042385551668019200000000000, 82086865668400428790437436119503664712777728000000000000000000

Offset: 1

Vaclav Kotesovec, Feb 20 2015

nonn

Eric Weisstein's World of Mathematics, Hyperfactorial
Eric Weisstein's World of Mathematics, Superfactorial

Cf. A000178, A002109, A036740, A055462, A255268.

Table[Product[k!^k,{k,1,n}],{n,1,10}]
FoldList[Times,Table[(k!)^k,{k,10}]] (* Harvey P. Dale, Aug 16 2021 *)

a(n) = A255268(n) / A055462(n-1).

a(n) ~ sqrt(A) * exp((3 - 45*n^2 - 32*n^3 - 9*Zeta(3)/Pi^2)/72) * n^((8*n^3 + 18*n^2 + 10*n + 1)/24) * (2*Pi)^(n*(n+1)/4), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... .

A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Alternatively, write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values. Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324].
If {(W,X,Y,Z)} are 4-tuples of nonnegative integers with W>=X>=Y>=Z ordered by W, X, Y and Z, then W=A127321(n), X=A127322(n), Y=A127323(n) and Z=A127324(n). These sequences are the four-dimensional analogs of the three-dimensional A056556, A056557 and A056558.
This is a 'Matryoshka doll' sequence with alpha=0 (cf. A055462 and A000332), seq(seq(seq(seq(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..4). - Peter Luschny, Jul 14 2009

			See A127321 for a table of A127321, A127322, A127323, A127324
See A127327 for a table of A127324, A127325, A127326, A127327

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A127321, A127322, A127323, A056556, A056557, A056558, A000332, A000292, A000217.

Cf. A002262, A056558.

import Data.List (inits)
a127324 n = a127324_list !! n
a127324_list = concatMap (concatMap concat .
               inits . inits . enumFromTo 0) $ enumFrom 0
-- Reinhard Zumkeller, Jun 01 2015

seq(seq(seq(seq(i,i=0..k),k=0..n),n=0..m),m=0..5); # Peter Luschny, Sep 22 2011

Table[i, {m, 0, 5}, {k, 0, m}, {j, 0, k}, {i, 0, j}] // Flatten  (* Robert G. Wilson v, Sep 27 2011 *)

For W>=X>=Y>=Z>=0, a(A000332(W+3)+A000292(X)+A000217(Y)+Z) = Z A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? 0 : A127323(n)==A127324(n) ? 0 : A127324(n)+1

A066120 Superduperprimorials: product of first n terms in Chernoff sequence.

Original entry on oeis.org

1, 2, 24, 8640, 653184000, 114069441024000000, 598216546007097937920000000000, 1601591599167888308924824752807936000000000000000

Offset: 0

Henry Bottomley, Dec 05 2001

First number with n distinct positive triangular exponents in its prime factorization. Subsequence of A025487. - Matthew Vandermast, Nov 05 2009; edited May 23 2012

			a(4) = 2*(2*(2*3))*(2*(2*3)*(2*3*5)) = 2*(2*6)*(2*6*30) = 2*12*360 = 8640.

Harry J. Smith, Table of n, a(n) for n = 0..17

Cf. A000040, A002110, A006939, A055462, A166475.

a(n) = { my(p=primes(n)); prod(i=1, #p, p[i]^((n - i + 1)*(n - i + 2)/2)) } \\ Harry J. Smith, Feb 01 2010

a(n) = Product_{i=1..n} prime(i)^((n-i+1)*(n-i+2)/2) = a(n-1)*A006939(n) = A066119(n-3, 4).

A163085 Product of first n swinging factorials (A056040).

Original entry on oeis.org

1, 1, 2, 12, 72, 2160, 43200, 6048000, 423360000, 266716800000, 67212633600000, 186313420339200000, 172153600393420800000, 2067909047925770649600000, 7097063852481244869427200000

Offset: 0

Peter Luschny, Jul 21 2009

nonn

With the definition of the Hankel transform as given by Luschny (see link) which uniquely determines the original sequence (provided that all determinants are not zero) this is also 1/ the Hankel determinant of 1/(n+1) (assuming (0,0)-based matrices).
a(2*n-1) is 1/determinant of the Hilbert matrix H(n) (A005249).
a(2*n) = A067689(n). - Peter Luschny, Sep 18 2012

Peter Luschny, SequenceTransformations

Cf. A056040, A163086, A055462, A000178.

a := proc(n) local i; mul(A056040(i),i=0..n) end;

a[0] = 1; a[n_] := a[n] = a[n-1]*n!/Floor[n/2]!^2; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 26 2013 *)

def A056040(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i) for i in (0..n))
[A056040(i) for i in (0..14)] # Peter Luschny, Sep 18 2012

A255268 a(n) = Product_{k=1..n} k!^n.

Original entry on oeis.org

1, 4, 1728, 6879707136, 49302469038676377600000, 237376313799769806328950291431424000000000000, 487929826521303413461947888047619993419888153407795494912000000000000000000000

Offset: 1

Vaclav Kotesovec, Feb 20 2015

nonn

Eric Weisstein's World of Mathematics, Barnes G-Function
Eric Weisstein's World of Mathematics, Superfactorial

Cf. A000178, A055462, A074962, A255269.

Table[Product[k!,{k,1,n}]^n,{n,1,10}]
Table[BarnesG[n+2]^n, {n, 1, 10}]

a(n) = A000178(n)^n.

a(n) ~ exp(1/12 + n/12 - n^2 - 3*n^3/4) * n^(5*n/12 + n^2 + n^3/2) * 2^(n/2 + n^2/2) * Pi^(n/2 + n^2/2) / A^n, where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962).

A306635 a(n) = Product_{k=1..n} BarnesG(2*k).

Original entry on oeis.org

1, 2, 576, 14332723200, 72474629486854275072000000, 482580045081719158086051946616717605601280000000000000

Offset: 1

Vaclav Kotesovec, Mar 02 2019

nonn

Next term is too long to be included.

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function

Cf. A055462, A168467, A255674, A296607, A306651, A324992.

Table[Product[BarnesG[2*k], {k, 1, n}], {n, 1, 8}]
Round[Table[2^(2*n^3/3 + n^2 - 5*n/3 - 2/3) * E^(n^3/2 + 3*n^2/4 + n/4 + 1/12 - 3*Zeta[3]/(16*Pi^2) + 2*PolyGamma[-3, n + 1] + Derivative[1, 0][Zeta][-2, n + 1/2] + 2*Derivative[1, 0][Zeta][-1, n + 1/2]) * Gamma[n]^(2*n - 7/4) * BarnesG[2*n]^(n + 1) / (Glaisher^(2*n + 3) * Pi^(n^2/2 + n + 1/2) * n^(n^2) * Gamma[2*n]^(n^2 + n - 3/4) * BarnesG[n]^2), {n, 1, 8}]] (* Vaclav Kotesovec, Mar 04 2019 *)

a(n) ~ c * 2^(2*n^3/3 + n^2/2 - n/4 - 3/8) * n^(2*n^3/3 - n/4) * Pi^(n^2/2 - 3/8) / (A^(n-2) * exp(11*n^3/9 - n/3 - Zeta(3)/(2*Pi^2) + 1/12)), where c = A255674^2 = 1.1446513373245340524595435844492841792576337833610236993... and A is the Glaisher-Kinkelin constant A074962.
a(n) ~ 2^(2*n^3/3 + n^2/2 - n/4 - 1/8) * n^(2*n^3/3 - n/4) * Pi^(n^2/2) / (A^n * exp(11*n^3/9 - n/3 - Zeta(3)/(16*Pi^2))), where A is the Glaisher-Kinkelin constant A074962.
a(n) = a(n-1)*A296607(n). - R. J. Mathar, Jul 24 2025

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A193430 Primes p such that p+1 is in A055462.

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A238265 Primes p such that p + 1 or p - 1 is in A055462.

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A000178 Superfactorials: product of first n factorials.

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A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

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A255269 a(n) = Product_{k=1..n} k!^k.

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A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

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A066120 Superduperprimorials: product of first n terms in Chernoff sequence.

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